A SHORT INTRODUCTION TO DE BRANGES–ROVNYAK

SPACES

DAN TIMOTIN

Contents

1. Introduction 12. Preliminaries 23. Introducing de Branges–Rovnyak spaces 3

3.1. Model spaces 33.2. Reproducing kernels 43.3. Contractively included subspaces 53.4. The complementary space 6

4. More about contractively included subspaces 85. Back to H(b) 9

5.1. Some properties of H(b); definition of Xb 95.2. Another representation of H(b) and Xb 105.3. The dichotomy extreme/nonextreme 11

6. The nonextreme case 117. The extreme case 138. H(b) as a model space 159. Further reading 17References 17

Abstract. The notes provide a short introduction to de Branges–Rovnyak

spaces. They cover some basic facts and are intended to give the reader a

taste of the theory, providing sufficient motivation to make it interesting.

1. Introduction

The purpose of these notes is to provide a short introduction to de Branges–Rovnyak spaces, that have been introduced in [6, 7], an area that has seen sig-nificant research activity in the last years. They are intended to give to a casualreader a taste of the theory, providing sufficient motivation and connections withother domains to make it, hopefully, interesting. There exist two comprehensivereferences on the subject: the older book of Sarason [22] that contains most of thebasic facts, and the more recent monograph of Fricain and Mashreghi [14]. Theinterested reader may study in depth the subject from there.

The author is partially supported by a grant of the Romanian National Authority for ScientificResearch, CNCS UEFISCDI, project number PN-II-ID-PCE-2011-3-0119.

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2 DAN TIMOTIN

The prerequisites are basic facts in operator theory on Hilbert space and in Hardyspaces. Many books contain them, but we have preferred to give as a comprehen-sive reference Nikolski’s treaty [19], where all can be found (see the beginning ofSection 2).

So the plan of the notes is the following. We introduce the de Branges–Rovnyakspaces as natural reproducing kernel spaces, generalizing the model spaces that ap-pear prominently in the theory of contractions. The challenge here is to justify ina sufficient manner a rather exotic object of study, namely contractively includedsubspaces; we have considered the reproducing kernel approach as especially con-venient. We give next some basic properties, following closely [22].

The dichotomy b extreme/nonextreme appears soon. The general idea is thatthe extreme case has many features that are not far from the case of b inner (theclassical model spaces), while the nonextreme case is more exotic from this pointof view.

Originally, the de Branges–Rovnyak spaces have been developed in view of modeltheory, that is, giving a universal model for certain class of operators on Hilbertspace. They have been in the shade for a few decades, as the much more popularand well developed model theory of Sz–Nagy and Foias [24] has gained the upperhand. It is known to experts that the two theories are equivalent, and we thoughtthat a justification of the study of de Branges–Rovnyak spaces should include somepresentation of their role as model spaces. It turns out that this is easier done forthe extreme case, and we have chosen to present this case at the end of the notes.

As noted above, the basic reference for de Branges–Rovnyak spaces, that hasbeen frequently used in these notes, is the book of Sarason [22]. Some simpleunproved results appear in the text as exercises; for the others, references areindicated in the text at the relevant places, occasionally with a hint of the proof.

This work will appear in the Proceedings Volume of the CRM Conference In-variant Subspaces of the Shift Operator, August 26–30, 2013. The author is gratefulto the organizers of the Conference for the support provided.

2. Preliminaries

A comprehensive reference for all facts in this section is [19], which has theadvantage to contain both the necessary prerequisites from function theory (PartA, chapters 1–3) and from operator theory (part C, chapter 1).

If H is a Hilbert space and H ′ ⊂ H a closed subspace, we will write PH′ forthe orthogonal projection onto H ′. The space of bounded linear operators fromH1 to H2 is denoted B(H1, H2); in case H1 = H2 = H we write just B(H). IfT ∈ B(H1, H2) is a contraction, we will denote by DT the selfadjoint operator(I − T ∗T )1/2 and by DT the closed subspace DTH1 = kerT⊥. Thus DT ⊂ H1 andDT∗ ⊂ H2. Obviously DT |DT is one-to-one.

Exercise 2.1. We have TDT = DT∗T . In particular, T maps DT into DT∗ .

We may also consider the domain and/or the range of DT to be DT ; by an abuseof notation all these operators will still be denoted by DT . Note that the adjointof DT : DT → H1 is DT : H1 → DT .

Exercise 2.2. If T ∈ B(H) is a contraction, H1 ⊂ H is a closed subset

invariant by T , and we denote T1 = T |H1, then DT1 = PH1DT .

A SHORT INTRODUCTION TO DE BRANGES–ROVNYAK SPACES 3

We denote by L2, L∞ the Lebesgue spaces on the unit circle T; we will alsomeet their closed subspaces H2 ⊂ L2 and H∞ ⊂ L∞ (the Hardy spaces). Thecorresponding norms will be denoted by ‖ · ‖2 and ‖ · ‖∞ respectively. As usual,H2 and H∞ can be identified with their analytic extension inside the unit discD. We assume known basic facts about inner and outer functions. We will writeP+ := PH2 (the orthogonal projection in L2). The symbols 0 and 1 will denote theconstant functions that take this value.

Each function φ in L∞ acts as multiplication on L2; the corresponding operatorwill be denoted by Mφ, and we have ‖Mφ‖ = ‖φ‖∞. In particular, if φ(z) = z, wewill write Z = Mφ. Actually, the commutant of Z (the class of all operators T onL2 such that ZT = TZ) coincides precisely with the class of all Mφ for φ ∈ L∞.(Obviously we have to define φ = T1 ∈ L2; a little work is needed to show that itis in L∞.)

The compression P+MφP+ restricted to the space H2 is called the Toeplitzoperator with symbol φ and is denoted by Tφ. Again we have ‖Tφ‖ = ‖φ‖∞;moreover, if φ ∈ H∞, then Tφ is one-to-one (this is a consequence of the brothersRiesz Theorem: a function in H2 is 6= 0 a.e.). In particular, if φ(z) = z, we willwrite S = Tφ; its adjoint S∗ acts as

(2.1) (S∗f)(z) =f(z)− f(0)

z.

We have T ∗φ = Tφ.As noted above, the multiplication operators commute; this is in general not

true for the Toeplitz operators.

Exercise 2.3. If φ ∈ H∞, or ψ ∈ H∞, then TψTφ = Tψφ.

If kλ(z) = 11−λz (a reproducing vector in H2—see more on reproducing kernels

in Subsection 3.2 below), then for any φ ∈ H∞ we have

(2.2) Tφkλ = φ(λ)kλ.

3. Introducing de Branges–Rovnyak spaces

3.1. Model spaces. Beurling’s theorem says that any subspace of H2 invariantby S is of the form uH2, with u an inner function.

Exercise 3.1. S|uH2 is unitarily equivalent to S.

From some points of view, the orthogonal Ku = H2 uH2 is more interesting:it is a model space. It is invariant by S∗, but S∗|Ku may behave very differently.Actually, we know exactly how differently :

Theorem 3.2. If T is a contraction on a Hilbert space H, then the following areequivalent:

(1) I − T ∗T and I − TT ∗ have rank one and Tn tends strongly to 0.(2) T is unitarily equivalent to Su := S∗|Ku ∈ B(Ku) for some inner function

u.

4 DAN TIMOTIN

Theorem 3.2 is a particular case of the general Sz.-Nagy–Foias theory of contrac-tions (see, for instance, the revised edition [24] of the original monography); onecan find it also in [19].

So Su is a model operator for a certain class of contractions. We will meet inthis course model operators for a more general class.

3.2. Reproducing kernels. We introduce a larger class of spaces that include Ku

for inner u; this will be done by means of reproducing kernels. A Hilbert space R offunctions on a set X is called a reproducing kernel space (RKS) if the evaluations atpoints of X are continuous; we will always have X = D. By Riesz’s representationtheorem it follows then that for each λ ∈ D there exists a function lRλ ∈ R, calledthe reproducing vector for λ, such that f(λ) = 〈f, lRλ 〉. The function of two variablesLR(z, λ) := lRλ (z) = 〈lRλ , lRz 〉 is called the reproducing kernel of the space R. Thereis a one-to-one correspondence between RKS’s and positive definite kernels (see forinstance [1]).

Exercise 3.3. (1) If R is a RKS, and R1 ⊂ R is a closed subspace,

then R1 is also a RKS, and lR1λ = PR1 l

Rλ .

(2) If R = R1 ⊕R2, then

(3.1) LR = LR1 ⊕ LR2 .

All three spaces discussed above have reproducing kernels, namely:

H2 −→ 11−λz ,

uH2 −→ u(λ)u(z)

1−λz ,

Ku −→ 1−u(λ)u(z)

1−λz ,

and one can check that equality (3.1) is satisfied.

Our plan is to obtain RKSs with similar formulas, but replacing the inner func-tion u with an arbitrary function b in the unit ball of H∞. That is, we want spaces

with kernels b(λ)b(z)

1−λz and 1−b(λ)b(z)

1−λz .

Of course it is not obvious that such RKSs exist. Then, if they exist, we wantto identify them concretely, hoping to relate them to the familiar space H2.

Things are simpler for the first kernel. Note first the next (general) exercise.

Exercise 3.4. If L(z, λ) is a positive kernel on X ×X and φ : X → C, thenφ(z)φ(λ)L(z, λ) is a positive kernel.

So b(λ)b(z)

1−λz is the kernel of some space R; but we would like to know it more

concretely. The case b = u inner suggests that a good candidate might be bH2.Now, we already have a problem: if b is a general function, bH2 might not be closedin H2, so it is not a genuine Hilbert space. But let us be brave and go on: we want

lRλ (z) = b(λ)b(z)kλ.

Since the reproducing kernel property should be valid in R, we must have, for anyf ∈ H2,

b(λ)f(λ) = 〈bf, lRλ (z)〉R = b(λ)〈bf, bkλ〉Rand therefore

f(λ) = 〈bf, bkλ〉R.

A SHORT INTRODUCTION TO DE BRANGES–ROVNYAK SPACES 5

On the other hand, since f ∈ H2, we have f(λ) = 〈f, kz〉H2 .We have now arrived at the crucial point. If b is inner, then 〈bf, bkz〉H2 =

〈f, kz〉H2 and everything is fine: the scalar product in bH2 is the usual scalarproduct in H2. But, in the general case, we have to define a different scalarproduct on R = bH2, by the formula

(3.2) 〈bf, bg〉R := 〈f, g〉H2 .

This appears to solve the problem. Since bf1 = bf2 implies f1 = f2, formula (3.2)is easily shown to define a scalar product on bH2. We will denote the correspondingHilbert space by M(b). Let us summarize the results obtained.

Theorem 3.5. M(b), defined as bH2 endowed with the scalar product (3.2), is aHilbert space, which as a set is a linear subspace (in general not closed) of H2. Its

reproducing kernel is b(λ)b(z)

1−λz , and the inclusion ι : M(b) → H2 is a contraction.

M(b) is invariant by S, and S acts as an isometry on M(b).

Proof. From (3.2) it follows that the map f 7→ bf is isometric from H2 ontoM(b),whenceM(b) is complete. The formula for the reproducing kernel has been proved(in fact, it lead to the definition of the space M(b)). We have

‖ι(bf)‖H2 = ‖bf‖H2 ≤ ‖f‖H2 = ‖bf‖M(b),

and thus ι is a contraction.Finally M(b) is invariant by S since z(bf) = b(zf), and

‖zbf‖M(b) = ‖b(zf)‖M(b) = ‖zf‖H2 = ‖f‖H2 = ‖bf‖M(b),

and thus the restriction of S is an isometry.

This settles the case of the kernel b(λ)b(z)

1−λz . To discuss 1−b(λ)b(z)

1−λz is slightly more

complicated, and a preliminary discussion is needed.

3.3. Contractively included subspaces. Let T : E → H be a bounded one-to-one operator. Define on the image T (E) a scalar product 〈·, ·〉′ by the formula

(3.3) 〈Tξ, Tη〉′ := 〈ξ, η〉E .

Then T is a unitary operator from E to T (E) endowed with 〈·, ·〉′. The space ob-tained is complete; we will denote it by M(T ). The linear space M(T ) is containedas a set in H, and the inclusion is a contraction if and only if T is a contraction.In this case the space M(T ) will be called a contractively included subspace of H,and the scalar product will be denoted 〈·, ·〉M(T ).

A slight modification is needed in case T is not one-to-one; then (3.3) cannotbe used directly since there Tξ does not determine ξ uniquely. We may howeverrecapture this uniqueness and use (3.3) if we require that ξ, η ∈ kerT⊥; then Tbecomes a unitary from kerT⊥ to M(T ).

In almost all cases in this course the corresponding contraction T will be one-to-one, and thus we will apply directly (3.3). The only exception appears in Theo-rem 5.3, when we will use Lemma 3.6 below, with no direct reference to the scalarproduct.

We have already met a particular case of this notion: with the above notations,we haveM(b) = M(Tb). Remember that, since b ∈ H∞, the operator Tb is one-to-one.

6 DAN TIMOTIN

The following is a basic result that is used when we have to deal with twocontractively embedded subspaces. It is essentially contained in [10].

Lemma 3.6. Suppose T1 : E1 → H, T2 : E2 → H are two contractions. Then:

(1) The space M(T1) is contained contractively in M(T2) (meaning that M(T1) ⊂M(T2) and the inclusion is a contraction) if and only if T1T

∗1 ≤ T2T

∗2 .

(2) The spaces M(T1) and M(T2) coincide as Hilbert spaces (that is, they areequal as sets, and the scalar product is the same) if and only if T1T

∗1 = T2T

∗2 .

(3) T : H → H acts contractively on M(T1) (meaning that T (M(T1)) ⊂M(T1)and T |M(T1) is a contraction) if and only if TT1T

∗1 T∗ ≤ T1T

∗1 .

It is worth at this point to note the following theorem of de Branges andRovnyak [6], which is an analogue of Beurling’s theorem.

Theorem 3.7. Suppose X ⊂ H2 is a contractively included subspace of H2. Thefollowing are equivalent

(1) X is invariant by S and the restriction S|X is an isometry (in the normof X).

(2) There exist a function b in the unit ball of H∞, such that X =M(b).

The function b is determined up to a multiplicative constant of modulus 1.

Suppose now that H is a reproducing kernel Hilbert space, with kernel L(z, λ).We may obtain for the reproducing vectors of M(T ) a formula similar to the par-ticular case from the previous subsection.

Lemma 3.8. Suppose T : E → H is one-to-one. With the above notations, wehave

lM(T )λ = TT ∗lHλ .

Proof. We have, using (3.3),

〈Tf, TT ∗lHλ 〉M(T ) = 〈f, T ∗lHλ 〉H = 〈Tf, lHλ 〉H = (Tf)(λ) = 〈Tf, lM(T )λ 〉M(T ),

which proves the theorem.

In case H = H2, T = Tb, we recapture the previous result: M(Tb) =M(b) and,using (2.2),

lM(b)λ = TbT

∗b kλ = bb(λ)kλ.

3.4. The complementary space. Remember that our current purpose is to find,

if possible, an RKS with kernel 1−b(λ)b(z)

1−λz . Let us note that

1− b(λ)b(z)

1− λz=

1

1− λz− b(λ)b(z)

1− λz= LH

2

(z, λ)− LM(b)(z, λ).

Can we obtain in the general case a formula for such a difference? The answer ispositive.

Lemma 3.9. With the above notations,

LH(z, λ)− LM(T )(z, λ) = LM(DT∗ )(z, λ),

where DT∗ : DT∗ → H.

A SHORT INTRODUCTION TO DE BRANGES–ROVNYAK SPACES 7

Proof. Using Lemma 3.8, we have

LH(z, λ)− LM(T )(z, λ) = 〈lHλ , lHz 〉H − 〈lM(T )λ , lM(T )

z 〉M(T )

= 〈lHλ , lHz 〉H − 〈TT ∗lH)λ , TT ∗lHz 〉M(T )

= 〈lHλ , lHz 〉H − 〈T ∗lH)λ , T ∗lHz 〉E

= 〈lHλ , (I − TT ∗)lHz 〉H = 〈DT∗ lHλ , DT∗ l

Hz 〉H

= 〈D2T∗ l

Hλ , D

2T∗ l

Hz 〉M(DT∗ ),

(the last equality being a consequence of the fact that DT∗ is one-to-one as anoperator from DT∗ into H). Then Lemma 3.8, applied to DT∗ instead of T , saysthat the last quantity is precisely LM(DT∗ )(z, λ).

Since DT∗ is a contraction, the RKS corresponding to LH(z, λ)−LM(T )(z, λ) isalso a space contractively included in H; it is called the space complementary toM(T ) and will be denoted by C(T ).

Exercise 3.10. If T is an isometry, then M(T ) is a usual subspace of H

(with the norm restricted), and C(T ) is its orthogonal complement.

If x ∈ H, then one can write

(3.4) x = TT ∗x+D2T∗x.

The first term in the right hand side is in M(T ), while the second is in C(T ).Moreover,

‖TT ∗x‖2M(T ) = ‖T ∗x‖2H , ‖D2T∗x‖2C(T ) = ‖DT∗x‖2H ,

whence‖x‖2 = ‖TT ∗x‖2M(T ) + ‖D2

T∗x‖2C(T ).

In case T is an isometry, M(T ) and C(T ) form an orthogonal decomposition ofH, and (3.4) is the corresponding orthogonal decomposition of H. In the generalcase M(T ) and C(T ) may have a nonzero intersection, and so a decompositionx = x1 + x2 with x1 ∈M(T ), x2 ∈ C(T ) is no more unique.

Exercise 3.11. If T is a contraction, and x = x1 + x2 with x1 ∈ M(T ),x2 ∈ C(T ), then

‖x‖2 ≤ ‖x1‖2M(T ) + ‖x2‖2C(T ),

and equality implies x1 = TT ∗x, x2 = D2T∗x.

At this point we have achieved our first purpose. Lemma 3.9 applied to thecase H = K = H2 and T = Tb yields the identification of the reproducing kernel

corresponding to 1−b(λ)b(z)

1−λz .

Theorem 3.12. The RKS with kernel 1−b(λ)b(z)

1−λz is M(DT∗b). It is a contrac-

tively included subspace of H2 that will be denoted H(b) and called the de Branges–Rovnyak space associated to the function b in the inner ball of H∞.

As noted above, if b = u is inner, then H(b) = Ku.

Exercise 3.13. (1) H2 = H(0) =M(1).(2) If ‖b‖∞ < 1, then H(b) is a renormed version of H2.(3) If infz∈D |b(z)| > 0, thenM(b) is a renormed version of H2.

8 DAN TIMOTIN

We will denote from now on kbλ = lH(b)λ = 1−b(λ)b(z)

1−λz .

4. More about contractively included subspaces

Lemma 4.1. If T : E → H is a contraction, then:

(1) ξ ∈ H belongs to C(T ) if and only if T ∗ξ ∈ C(T ∗).(2) If ξ1, ξ2 ∈ C(T ), then

〈ξ1, ξ2〉C(T ) = 〈ξ1, ξ2〉H + 〈T ∗ξ1, T ∗ξ2〉C(T∗).

Proof. The inclusion T ∗(C(T )) ⊂ C(T ∗) follows from the intertwining relation inExercise 2.1. On the other hand, if T ∗ξ ∈ C(T ∗), we have T ∗ξ = DT η for someη ∈ H, and, using again Exercise 2.1,

ξ = TT ∗ξ +D2T∗ξ = TDT η +D2

T∗ξ = DT∗(DT∗ξ + Tη),

which shows that ξ ∈ C(T ).To prove (2), write ξ1 = DT∗η1, ξ2 = DT∗η2, with η1, η2 ∈ DT∗ ; then T ∗η1, T

∗η2 ∈DT . Since both DT : DT → H and DT∗ : DT∗ → H are one-to-one, we have, us-ing (3.3) and Exercise 2.1,

〈ξ1, ξ2〉C(T ) = 〈η1, η2〉 = 〈DT∗η1, DT∗η2〉+ 〈T ∗η1, T∗η2〉

= 〈ξ1, ξ2〉+ 〈DTT∗η1, DTT

∗η2〉C(T∗)

= 〈ξ1, ξ2〉+ 〈T ∗DT∗η1, T∗DT∗η2〉C(T∗)

= 〈ξ1, ξ2〉+ 〈T ∗ξ1, T ∗ξ2〉C(T∗).

There is a more direct way in which complementarity is related to orthogonality.If T ∈ B(E,H) is a contraction, we define the Julia operator J(T ) : E ⊕ DT∗ →H ⊕DT by

J(T ) =

(T DT∗

DT −T ∗).

Exercise 4.2. The Julia operator is unitary.

Lemma 4.3. Suppose T : E → H is one-to-one. Denote

X1 = J(T )(E ⊕ 0), X2 = (H ⊕DT )X1 = J(T )(0 ⊕ DT∗),and by P1 the projection of H ⊕ DT onto its first coordinate H. Then P1|X1 isunitary from X1 onto M(T ), and P1|X2 is unitary from X2 onto C(T ).

Proof. We have

P1X1 = P1(Tx⊕DTx : x ∈ E) = Tx : x ∈ E = M(T ).

Moreover, if x1 ∈ X1, then x1 = Tx⊕DTx for some x ∈ E, and

‖P1x1‖M(T ) = ‖Tx‖M(T ) = ‖x‖E = ‖J(T )(x⊕ 0)‖ = ‖x1‖,which proves the first part of the lemma.

Now X2 = J(T )(0 ⊕ DT∗), so

P1X2 = P1(DT∗y ⊕−T ∗y : y ∈ DT∗) = DT∗y : y ∈ DT∗ = C(T ).

If x2 ∈ X2, then x2 = DT∗y ⊕−T ∗y for some y ∈ DT∗ , and

‖P1x2‖C(T ) = ‖DT∗y‖C(T ) = ‖y‖DT∗ = ‖J(T )(0⊕ y)‖ = ‖x2‖,as required.

A SHORT INTRODUCTION TO DE BRANGES–ROVNYAK SPACES 9

We can view this result as saying that the orthogonal decomposition of H⊕DT asX1 ⊕X2 is mapped by projecting onto the first coordinate into the complementarydecomposition H = M(T ) + C(T ) (which is not, in general, a direct sum). So therather exotic definition of complementary spaces is in fact the projection of a morefamiliar geometric structure.

5. Back to H(b)

5.1. Some properties of H(b); definition of Xb. We denote H(b) := H(Tb).Although our focus is on H(b), the space H(b) is a useful tool for its study.

Lemma 5.1. H(b) is contained contractively in H(b).

Proof. We have

TbTb = P+MbP+MbP+|H2 ≤ P+MbMbP+|H2

= P+MbMbP+|H2 = P+MbP+MbP+|H2 = TbTb.

Therefore

D2Tb≤ D2

T∗b,

whence Lemma 3.6(1) implies that H(b) is contained contractively in H(b).

Lemma 4.1 applied to the case T = Tb yields the following result.

Lemma 5.2. If h ∈ H2, then h ∈ H(b) if and only if Tbh ∈ H(b). If h1, h2 ∈ H(b),then

(5.1) 〈h1, h2〉H(b) = 〈h1, h2〉H2 + 〈Tbh1, Tbh2〉H(b).

We now show that, similarly to model spaces, de Branges–Rovnyak spaces areinvariant by adjoints of Toeplitz operators.

Theorem 5.3. If φ ∈ H∞, then H(b) and H(b) are both invariant under T ∗φ = Tφ,

and the norm of this operator in each of these spaces is at most ‖φ‖∞.

Proof. We may assume that ‖φ‖∞ ≤ 1. By Lemma 3.6(3), in order to show thatTφ acts as a contraction in H(b) we have to prove the inequality

Tφ(I − TbTb)Tφ ≤ I − TbTb,

or

0 ≤ I − TbTb − Tφ(I − TbTb)Tφ = I − TbTb − TφTφ + TφTbTbTφ

= I − T|b|2 − T|φ|2 + T|b|2|φ|2 = T(1−|b|2)(1−|φ|2).

But the last operator is the compression to H2 of M(1−|b|2)(1−|φ|2), which is positive,

since (1− |b|2)(1− |φ|2) ≥ 0.This proves the statement for H(b). Take now h ∈ H(b). Lemma 5.2 implies

that Tbh ∈ H(b). By what has been just proved, TbTφh = TφTbh ∈ H(b), and thenapplying again Lemma 5.2 we obtain Tbh ∈ H(b).

Finally, using (5.1) and the contractivity of Tφ on H2 as well as on H(b), wehave

‖Tφh‖2H(b) = ‖Tφh‖2H2 + ‖TφTbh‖2H(b) ≤ ‖h‖2H2 + ‖Tbh‖2H(b) = ‖h‖2H(b),

so Tφ acts as a contraction in H(b).

10 DAN TIMOTIN

The most important case is obtained when φ(z) = z. Theorem 5.3 says thenthat H(b) is invariant under S∗ and the restriction of S∗ is a contraction. We willdenote by Xb this restriction S∗|H(b).

Corollary 5.4. The function S∗b is in H(b).

Proof. We have

TbS∗b = S∗Tbb = S∗TbTb1 = −S∗(I − TbTb)1

(for the last equality we have used the fact that S∗1 = 0). Obviously (I −TbTb)1 ∈H(b), so Theorem 5.3 implies that TbS

∗b = −S∗(I−TbTb)1 ∈ H(b). By Lemma 5.2it follows that S∗b ∈ H(b).

Note that if b 6= 1, then S∗b 6= 0. Besides S∗b, we know as inhabitants ofH(b) the reproducing vectors kbλ. Other elements may be obtained, for instance, byapplying to these elements powers or functions of Xb.

Exercise 5.5. Show that, if λ ∈ D, then

((I − λXb)−1(S∗b))(z) =b(z)− b(λ)z − λ .

Therefore the functions in the right hand side belong to H(b).

In general b itself may not be inH(b); we will see later exactly when this happens.Let us also compute the adjoint of Xb.

Lemma 5.6. If h ∈ H(b), then

X∗b h = Sh− 〈h, S∗b〉H(b)b.

Proof. A computation shows that Xbkbλ = S∗kbλ = λkbλ−b(λ)S∗b. Then, if h ∈ H(b)

and λ ∈ D, then

(X∗b h)(λ) = 〈X∗b h, kbλ〉H(b) = 〈h,Xbkbλ〉H(b) = λ〈h, kbλ〉H(b) − b(λ)〈h, S∗b〉H(b)

= λh(λ)− 〈h, S∗b〉H(b)b(λ),

which proves the lemma.

5.2. Another representation of H(b) and Xb. In the sequel of the course we will

use the notation ∆ = (1− |b|2)1/2. The spaces ∆H2 and ∆L2 are closed subspacesof L2 invariant with respect to Z. We will denote by V∆ and Z∆ the correspondingrestrictions of Z.

Exercise 5.7. V∆ is isometric, while Z∆ is unitary.

The next result, a slight modification of Lemma 4.3, provides another represen-tation of H(b).

Theorem 5.8. Suppose that b is a function in the unit ball of H∞. Then:

(1) S ⊕ V∆ is an isometry on H2 ⊕∆H2.

(2) The space Kb := (H2 ⊕ ∆H2) bh ⊕ ∆h : h ∈ H2 is a subspace of

H2 ⊕∆H2 invariant with respect to S∗ ⊕ V ∗∆.

(3) The projection P1 : H2 ⊕ ∆H2 → H2 on the first coordinate maps Kbunitarily onto H(b), and P1(S∗ ⊕ V ∗∆) = XbP1.

A SHORT INTRODUCTION TO DE BRANGES–ROVNYAK SPACES 11

Proof. The proof of (1) is immediate. Also, the map h 7→ bh ⊕∆h is an isometryof H2 onto bh ⊕ ∆h : h ∈ H2, which is therefore a closed subspace. Since(S ⊕ V∆)(bh ⊕∆h) = b(zh) ⊕∆(zh), it is immediate that bh ⊕∆h : h ∈ H2 isinvariant by S ⊕ V∆, whence its orthogonal K is invariant by S∗ ⊕ V ∗∆; thus (2) isproved.

To prove (3), let us apply Lemma 4.3 to the case T = Tb, when C(T ) = H(b).It says that, if X2 = (H2⊕DTb

)Tbh⊕DTbh, then the projection onto the first

coordinate maps X2 unitarily onto H(b). Since, for any h ∈ H2,

‖DTbh‖2 = ‖h‖2 − ‖bh‖2 =

1

2π

∫ π

−π|h(eit)|2 dt− 1

2π

∫ π

−π|b(eit)h(eit)|2 dt = ‖∆h‖2,

the map DTbh 7→ ∆h extends to a unitary U from DTb

onto the closure of ∆H2.

Then IH2⊕U maps unitarily H2⊕DTbonto H2⊕∆H2, X1 onto bh⊕∆h : h ∈ H2,

X2 onto Kb, and it commutes with the projection on the first coordinate. ThereforeP1 maps Kb unitarily onto H(b), and

P1(S∗ ⊕ V ∗∆)|Kb = P1(S∗ ⊕ V ∗∆)P1|Kb = (S∗ ⊕ 0)P1|Kb = (S∗|H(b))P1 = XbP1.

5.3. The dichotomy extreme/nonextreme. The study of the spacesH(b) splitsfurther into two mutually exclusive cases: when b is an extreme point of the unitball of H∞ and when it is not. The first case includes b = u inner, and thus will bemore closely related to model spaces, while the second includes the case ‖b‖ < 1,and thus there will be properties similar to the whole of H2. Actually, we will notuse extremality directly, but rather through one of the equivalent characterizationsgiven by the next lemma (for which again [19] can be used as a reference).

Lemma 5.9. If h is a function in the unit ball of H∞, then the following areequivalent:

(1) b is extreme.(2) 1

2π

∫ π−π log ∆(eit) dt = −∞.

(3) ∆H2 = ∆L2.

6. The nonextreme case

When 12π

∫ π−π log ∆(eit) dt > −∞, there exists a uniquely defined outer function

a with |a| = ∆ and a(0) > 0; thus |a|2 + |b|2 = 1. This function is a basic tool inthe theory of H(b) in the nonextreme case. Since (see Exercise 2.3)

TaTa = Taa = I − Tbb = I − TbTbLemma 3.6(2) implies that H(b) =M(a).

Exercise 6.1. If a is an outer function, then kerTa = 0.

We can apply Lemma 5.2 to the current situation.

Lemma 6.2. (1) We have h ∈ H(b) if and only if Tbh ∈ M(a); when thishappens there is a unique (by Exercise 6.1) function h+ ∈ H2 such thatTbh = Tah

+.(2) If h1, h2 ∈ H(b), then

〈h1, h2〉H(b) = 〈h1, h2〉H2 + 〈h+1 , h

+2 〉H2 .

(3) If h ∈ H(b) and φ ∈ H∞, then (Tφh)+ = Tφh+.

12 DAN TIMOTIN

Proof. (1) is a consequence of Lemma 5.2 and the equality H(b) = M(a); theuniqueness of h+ follows from Exercise 6.1.

The formula for the scalar product in Lemma 5.2 becomes

〈h1, h2〉H(b) = 〈h1, h2〉H2 + 〈Tah+1 , Tah

+2 〉H(a) = 〈h+

1 , h+2 〉H2 ,

the last equality being a consequence of the fact that Ta is one-to-one. This proves(2).

Finally,TbTφh = TφTbh = TφTah

+ = TaTφh+,

proving (3).

We gather in a theorem some properties of H(b) for b nonextreme.

Theorem 6.3. Suppose b is nonextreme.

(1) The polynomials belong to M(a) and are dense in M(a).(2) M(a) is dense in H(b).(3) The polynomials are dense in H(b).(4) The function b is in H(b), and ‖b‖2H(b) = |a(0)|−2 − 1.

(5) The space H(b) is invariant by the unilateral shift S.

Proof. By checking the action on monomials, it is immediate that the space Pn ofpolynomials of degree less or equal to n is invariant by Ta. But Ta is one-to-one,and so Ta|Pn is also onto. So all polynomials belong to the image of Ta, which isM(a). Moreover, since Ta is one-to-one, it is unitary as an operator from H2 toM(a). Then the image of all polynomials, which form a dense set in H2, is a denseset in M(a) which proves (1).

Suppose that h ∈ H(b) is orthogonal in H(b) to all M(a). In particular, h isorthogonal to TaS

∗nh for every n ≥ 0. By Lemma 6.2(3), (TaS∗nh)+ = TaS

∗nh+;applying then 6.2(2), we have, for any n ≥ 0,

0 = 〈h, TaS∗nh〉H(b)

= 〈h, TaS∗nh〉H2 + 〈h+, TaS∗nh+〉H2

=1

2π

∫ π

−πa(eit)(|h(eit)|2 + |h+(eit)|2)eint dt.

Therefore, the function a(|h|2 + |h+|2) belongs to H10 . A classical fact about outer

functions (see, for instance, [19]) implies that we also have |h|2 + |h+|2 ∈ H10 . But

the only real-valued function in H10 is the zero function, so h = 0, which proves (2).

Obviously, (1) and (2) imply (3).We have

Tbb = P+(bb) = P+(1− aa) = Ta(1/a(0)− a).

By Lemma 6.2 it follows that b ∈ H(b) and b+ = 1/a(0)− a; moreover,

‖b‖2H(b) = ‖b‖2H2 + ‖1/a(0)− a‖2H2

= ‖b‖2H2 + ‖a− a(0)‖2H2 + ‖1/a(0)− a(0)‖2H2

= ‖b‖2H2 + ‖a‖2H2 − |a(0)|2 + |a(0)|−2 + |a(0)|2 − 2

= |a(0)|−2 − 1,

which proves (4).Finally, Lemma 5.6 together with (4) prove the invariance of H(b) to S.

A SHORT INTRODUCTION TO DE BRANGES–ROVNYAK SPACES 13

7. The extreme case

We point out first some differences with respect to the nonextreme case.

Theorem 7.1. Suppose b is extreme. Then:

(1) The function b does not belong to H(b).(2) If b 6= 1, then H(b) is not invariant by S.

Proof. Suppose b ∈ H(b). By Theorem 5.8, it follows that there exists ψ ∈ ∆H2 ⊂L2, such that b⊕ ψ ⊥ bh⊕∆h : h ∈ H2. So

〈b⊕ ψ, bh⊕∆h〉 = 0, for all h ∈ H2,

which is equivalent to |b|2 + ∆ψ ∈ H20 . This is equivalent to 1 −∆2 + ∆ψ ∈ H2

0 ,whence f := ∆2 − ∆ψ is a nonzero (note that its zeroth Fourier coefficient is 1)function in H2. Thus ∆−1f = ∆− ψ ∈ L2, or ∆−2|f |2 ∈ L1.

We assert that this is not possible. Indeed,

∆−2|f |2 ≥ log(∆−2|f |2) = 2 log |f | − 2 log ∆.

Integrating, we obtain

1

2π

∫∆−2(eit)|f(eit)|2 dt ≥ 2

1

2π

∫log |f(eit)| dt+ 2

1

2π

∫(− log ∆(eit)) dt,

which cannot be true, since the first two integrals are finite, while the third isinfinite by Lemma 5.9.

We have thus proved (1). Then (2) follows from Lemma 5.6, which can berestated as

〈h, S∗bH(b)〉b = Sh−X∗b h.

So, if we choose h not orthogonal (in H(b)) to S∗b (in particular, h = S∗b), wemust have Sh /∈ H(b).

In the sequel we will use the geometrical representation of H(b) given by Theo-

rem 5.8. Using Lemma 5.9, we may replace in its statement ∆H2 by ∆L2.Denote b(z) = b(z), and ∆ = (1−|b|2)1/2. The map f 7→ f is a unitary involution

that maps L2 onto L2, H2 onto H2, and ∆L2 onto ∆L2.

Exercise 7.2. b is extreme if and only if b is extreme.

Theorem 7.3. Suppose b is extreme. Define

Ω : L2 ⊕ ∆L2 → L2 ⊕∆L2

by the formula

Ω(f ⊕ g) =(b(z)zf(z) + ∆zg(z)

)⊕(

∆zf(z)− bzg(z)).

Then Ω is unitary, it maps Kb onto Kb, and

(7.1) ΩXb = X∗bΩ.

14 DAN TIMOTIN

Proof. Ω acts as the unitary f ⊕ g 7→ zf(z)⊕ zg(z) followed by the unitary J(Mb),so it is unitary. We have

Kb = (L2 ⊕∆L2)[(H2− ⊕ 0)⊕ (bf ⊕∆f : f ∈ H2)

],

Kb = (L2 ⊕ ∆L2)[(H2− ⊕ 0)⊕ (bf ⊕ ∆f : f ∈ H2)

].

If f ∈ H2−, then

Ω(f ⊕ 0) = bzf(z)⊕∆zf(z).

But the map f 7→ zf(z) is a unitary from H2− onto H2, whence it follows that

Ω(H2−⊕0) = bf ⊕∆f : f ∈ H2. Similarly we obtain Ω(bf ⊕ ∆f : f ∈ H2) =

H2− ⊕ 0. Therefore

Ω((H2− ⊕ 0)⊕ (bf ⊕ ∆f : f ∈ H2)) = (H2

− ⊕ 0)⊕ (bf ⊕∆f : f ∈ H2),

whence Ω(Kb) = Ω(Kb).Finally, we have Xb = PKb

(Z∗ ⊕ Z∗∆)PKb|Kb and Xb = PKb

(Z∗ ⊕ Z∗∆

)PKb|Kb.

But Ω(Kb) = Ω(Kb) implies PKb= Ω∗PKb

Ω. Therefore

ΩXb = ΩPKb(Z∗ ⊕ Z∗

∆)PKb

|Kb = ΩΩ∗PKbΩ(Z∗ ⊕ Z∗

∆)Ω∗PKb

Ω|Kb= PKb

Ω(Z∗ ⊕ Z∗∆

)Ω∗PKbΩ|Kb.

But one checks easily that Ω(Z∗ ⊕ Z∗∆

)Ω∗ = Z ⊕ Z∆, and therefore

PKbΩ(Z∗ ⊕ Z∗

∆)Ω∗PKb

Ω|Kb = PKb(Z ⊕ Z∆)PKb

Ω|Kb= (PKb

(Z∗ ⊕ Z∗∆)PKb)∗Ω|Kb = X∗bΩ,

which ends the proof of the theorem.

We note that in case b is nonextreme X∗b is never unitarily equivalent to Xb.

Theorem 7.4. Suppose b is extreme. Then dimDXb= dimDX∗b = 1, and there

exists no subspace of Hb invariant by X and such that its restriction therein is anisometry.

Proof. From Theorem 5.8(3) it follows that we may prove the properties for therestriction S∗ ⊕ Z∗∆|Kb. Since S∗ acts isometrically on H2

0 and Z∗∆ is unitary, wehave DS∗⊕Z∗∆ = C(1⊕0). By Exercise 2.2, DXb

= CPKb(1⊕0). But PKb

(1⊕0) 6= 0;

indeed, otherwise we would have 1⊕0 = bh⊕∆h for some h ∈ H2; since h 6= 0 a.e.,this would imply ∆ = 0 a.e., or b inner, which is impossible if 1 = bh. ThereforedimDX = 1.

Applying the same argument to b and using (7.1), it follows that dimDX∗ = 1.Finally, suppose Y ⊂ K is a closed subspace on which Xb acts isometrically, and

h⊕ g ∈ Y, we have, for any n ≥ 0,

‖h‖2+‖g‖2 = ‖h⊕g‖2 = ‖S∗nh⊕Z∗∆ng‖2 = ‖S∗hn‖2+‖Z∗∆ng‖2 → ‖Z∗∆ng‖2 = ‖g‖2,

whence h = 0. But then we must have 0 ⊕ g ⊥ bf ⊕ ∆f for all f ∈ H2, org ⊥ ∆H2 = ∆L2. Since, on the other hand, g ∈ ∆L2, it follows that g = 0, whichends the proof of the theorem.

A SHORT INTRODUCTION TO DE BRANGES–ROVNYAK SPACES 15

8. H(b) as a model space

The purpose of this section is to prove the converse of Theorem 7.4. This willshow that in the extreme case the de Branges–Rovnyak spaces are model spacesfor a large class of operators. The theorem below (as well as its proof) is in facta particular case of the much more general analysis of contractions done in theSz.Nagy–Foias theory (see [24]). Here we have adapted the argument to a “minimal”self-contained form.

Theorem 8.1. Suppose T ∈ B(H) is a contraction such that dimDT = dimDT∗ =1, and there exists no subspace of H invariant by T and such that its restrictiontherein is an isometry. Then there exists an extreme b in the unit ball of H∞ suchthat T is unitarily equivalent to Xb.

Proof. Since the proof is rather long, we divide it in several steps.

Step 1. Dilation of T . To find the required function b, we will develop a certaingeometrical construction. Changing the order of the components in the range of theJulia operator yields a unitary operator mapping H ⊕DT∗ into DT ⊕H according

to the matrix(DT −T∗T DT∗

). We can extend this unitary to a unitary W acting on the

single enlarged space

H = · · · ⊕ DT∗ ⊕DT∗ ⊕H ⊕DT ⊕DT ⊕ . . . ,

that can be written as an bi-infinite operator matrix:

(8.1) W =

. . .

11

DT −T ∗

T DT∗

11

. . .

where the boxed entry corresponds to the central entry T : H → H. If we writeH = H− ⊕H ⊕ H+, with

H− = · · · ⊕ DT∗ ⊕DT∗ , H+ = DT ⊕DT ⊕ . . . ,

then H− is invariant by W , which acts therein as translation to the left, whileH+ is invariant by W ∗, whose restriction is translation to the right. (This is aconsequence of the fact that the 1 entries in the definition of W are all locatedimmediately above the main diagonal.)

Step 2. Two embeddings of L2 into H. Take a unit vector ε−−1 in the DT∗ com-

ponent of H− which is mostly to the right, and define, for n ∈ Z, ε−n = W−n−1ε−−1.Since dimDT∗ = 1, the family (ε−n )n≤−1 forms an orthonormal basis of H−. More-over, the whole family (ε−n )n∈Z is an orthonormal set in H (exercise!).

As (eint)n∈Z is an orthonormal basis in L2, we may define ω− : L2 → H to be theunique isometry that satisfies ω−(eint) = ε−n for all n ∈ Z. One checks easily thatits image ω−(L2) is a reducing space for W , and ω∗−Wω− = Me−it . The orthogonalprojection onto ω−(L2) is ω−ω

∗−, and it commutes with W .

16 DAN TIMOTIN

An analogous construction can be made for H+. We obtain an orthonormal set(ε+n )n∈Z in H, such that (ε+n )n≥0 is a basis for H+. Then ω+ : L2 → H is theisometry that satisfies ω+(eint) = ε+n for all n ∈ Z; ω+(L2) is also a reducing spacefor W , ω∗+Wω+ = Me−it , and ω+ω

∗+W = Wω+ω

∗+.

Step 3. Finding b. Consider then the map ω∗−ω+ : L2 → L2. We have, using theabove remarks as well as the equalities ω∗+ω+ = ω∗−ω− = IL2 ,

ω∗−ω+Me−it = ω∗−ω+ω∗+Wω+ = ω∗−Wω+ω

∗+ω+ = ω∗−Wω+

= (ω∗−ω−)ω∗−Wω+ = ω∗−Wω−ω∗−ω+ = Me−itω∗−ω+.

(8.2)

So ω∗−ω+ commutes with Me−it ; it follows that it commutes also with its inverseMeit (exercise!). We have noticed in the introduction that in this case we musthave ω∗−ω+ = Mb for some function b ∈ L∞, and ‖ω∗−ω+‖ ≤ 1 implies ‖b‖∞ ≤ 1.

Now, b = Mb1 = ω∗−ω+1 = ω∗−ε+0 . Since ε+0 ∈ H+ ⊥ H− = ω−(H2

0 ), it follows

that ω∗−ε+0 ∈ H2. Thus b ∈ L∞ ∩H2 = H∞, and we have found our candidate for

the function in the unit ball of H∞. It remains now to check that it satisfies therequired properties. As above, we will denote ∆ = (1− |b|2)1/2 ∈ L∞.

Step 4. Constructing the unitary equivalence. Let us now note that theclosed linear span ω+L

2 ∨ ω−L2 equals H. Indeed, it reduces W and contains H+

and H−; thus its orthogonal Y has to be a reducing subspace of W contained in H(more precisely, in its embedding in H). From (8.1) it follows then that W |Y = T |Y ,so Y should be a subspace of H invariant by T and such that the restriction isisometric (even unitary!), which contradicts the hypothesis. Thus Y = 0.

We define then a mapping U : ω+L2 ∨ ω−L2 → L2 ⊕∆L2 by

U(ω+f+ + ω−f−) = (f− + bf+)⊕∆f+.

We have

‖ω+f+ + ω−f−‖2 = ‖ω+f+‖2 + ‖ω−f−‖2 + 2<〈ω+f+, ω−f−〉= ‖f+‖22 + ‖f−‖22 + 2<〈ω∗−ω+f+, f−〉2 = ‖f+‖22 + ‖f−‖22 + 2<〈bf+, f−〉2,

and

‖(f− + bf+)⊕∆f+‖2 =

∫|f− + bf+|2 +

∫∆2|f+|2

=

∫|f−|2 + |bf+|2 + 2<bf+f− + ∆2|f+|2

=

∫|f−|2 + |f+|2 + 2<bf+f−,

whence U is an isometry, and it is easy to see that the image is dense. It can beextended to a unitary operator, that we will denote by the same letter,

U : H→ L2 ⊕∆L2.

The commutation relations satisfied by ω± imply that UW = (Z∗ ⊕ Z∗∆)U .

Step 5. Final checks. Now U(H) = U(H−)⊕ U(H)⊕ U(H+), and

U(H−) = U(ω−(H20 )) = f− ⊕ 0 : f− ∈ H2

0 = H20 ⊕ 0,

U(H+) = U(ω+(H2)) = bf+ ⊕∆f+ : f+ ∈ H2,

A SHORT INTRODUCTION TO DE BRANGES–ROVNYAK SPACES 17

so

U(H) = (L2 ⊕∆L2)(

(H20 ⊕ 0)⊕ (bf+ ⊕∆f+ : f+ ∈ H2)

)= (H2 ⊕∆L2) (bf+ ⊕∆f+ : f+ ∈ H2).

As shown by (8.1), T can be viewed as the compression of W to H, so it is unitarilyequivalent through U to the compression of Z∗ ⊕ Z∗∆ to U(H). This last is easily

seen to be the restriction of S∗ ⊕ Z∗∆ to (H2 ⊕∆L2) (bf+ ⊕∆f+ : f+ ∈ H2).We are very close to the end: Theorem 5.8 would end the proof, provided we

could replace in the formula for U(H) the space H2 ⊕ ∆L2 with H2 ⊕ ∆H2 andthus Z∆ with V∆. But the space

Y := (H2 ⊕∆L2) (H2 ⊕∆H2) = 0 ⊕ (∆L2 ∆H2)

is invariant with respect to S∗⊕Z∗2 , which acts on it isometrically. By assumption,

we must have Y = 0, which means that H2 ⊕ ∆L2 = H2 ⊕ ∆H2; this finishesthe proof.

Exercise 8.2. Show that Theorem 8.1 implies Theorem 3.2.

9. Further reading

We discuss in this section some directions in which the study of de Branges–Rovnyak spaces has developed.

The model spaces Ku have no nonconstant multipliers. However, the theory ofmultipliers is interesting for the case of de Branges–Rovnyak spaces correspondingto nonextreme b; see references [16, 17, 18].

Integral representations of de Branges–Rovnyak spaces appear in [4], and havefurther been developed in [21, 12, 13]. The last paper is used in [5] to obtainweighted norm inequalities for functions in de Branges–Rovnyak spaces.

The connection between de Branges–Rovnyak and Dirichlet spaces is exploitedin [8, 9]; that between de Branges–Rovnyak spaces and composition operatorsin [15].

Finally, it is natural from many points of view (including that of model spaces)to consider also matrix or operator valued de Branges–Rovnyak spaces. These havealready been introduced in [4]; see [2, 3] for some recent developments.

References

[1] N. Aronszajn, Theory of reproducing kernels, Trans. AMS 68 (1950), 337–404.[2] J.A. Ball, V. Bolotnikov, S. ter Horst, Interpolation in de Branges-Rovnyak spaces, Proc.

Amer. Math. Soc. 139 (2011), 609–618.

[3] J.A. Ball, V. Bolotnikov, S. ter Horst, Abstract interpolation in vector-valued de Branges-Rovnyak spaces, Integral Equations Operator Theory 70 (2011), 227–263.

[4] J.A. Ball, Th.L. Kriete, Operator-valued Nevanlinna-Pick kernels and the functional modelsfor contraction operators, Integral Equations Operator Theory 10 (1987), 17–61.

[5] A. Baranov, E. Fricain, J. Mashreghi, Weighted norm inequalities for de Branges-Rovnyakspaces and their applications, Amer. J. Math. 132 (2010), 125–155.

[6] L. de Branges, J. Rovnyak, Square Summable Power Series, Holt, Rinehart and Winston,1966.

[7] L. de Branges, J. Rovnyak, Canonical models in quantum scattering theory, in PerturbationTheory and its Applications in Quantum Mechanics, ed. by C.H. Wilcox, Wiley, 1966, 295–392.

18 DAN TIMOTIN

[8] N. Chevrot, D. Guillot, Th. Ransford, De Branges-Rovnyak spaces and Dirichlet spaces, J.

Funct. Anal. 259 (2010), 2366–2383.

[9] C. Costara, Th. Ransford, Which de Branges–Rovnyak spaces are Dirichlet spaces (and viceversa)?, J. Funct. Anal. 265 (2013), 3204–3218.

[10] R.G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert

space, Proc. AMS 17 (1966), 413–415.[11] E. Fricain, Bases of reproducing kernels in de Branges spaces, J. Funct. Anal. 226 (2005),

373–405.

[12] E. Fricain, J. Mashreghi, Boundary behavior of functions in the de Branges-Rovnyak spaces,Complex Anal. Oper. Theory 2 (2008), 87–97.

[13] E. Fricain, J. Mashreghi, Integral representation of the n-th derivative in de Branges-Rovnyak

spaces and the norm convergence of its reproducing kernel, Ann. Inst. Fourier (Grenoble)58 (2008), 2113–2135.

[14] Emmanuel Fricain, Javad Mashreghi: Theory of H(b) Spaces, vol. I and II, New Monographsin Mathematics, Cambridge University Press, to appear.

[15] M.T. Jury, Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted compo-

sition operators, Proc. Amer. Math. Soc. 135 (2007), 3669–3675.[16] B.A. Lotto, D. Sarason, Multiplicative structure of de Branges’s spaces, Rev. Mat. Iberoamer-

icana 7 (1991), 183–220.

[17] B.A. Lotto, D. Sarason, Multipliers of de Branges-Rovnyak spaces, Indiana Univ. Math. J.42 (1993), 907–920.

[18] B.A. Lotto, D. Sarason, Multipliers of de Branges-Rovnyak spaces, II, in Harmonic Analysis

and Hypergroups, Delhi, 1995, 51–58.[19] N.K. Nikolski: Operators, Functions, and Systems: An Easy Reading, AMS, 2002.

[20] N.K. Nikolski, V.I. Vasyunin, Notes on two function models, in The Bieberbach conjecture

(West Lafayette, Ind., 1985), , Math. Surveys Monogr., 21, Amer. Math. Soc., Providence,RI, 1986, 113–141.

[21] A.G. Poltoratskii, Boundary behavior of pseudocontinuable functions, Algebra i Analiz 5(1993), 189–210, translation in St. Petersburg Math. J. 5 (1994), no. 2, 389–406.

[22] D. Sarason, Sub-Hardy Hilbert Spaces in the Unit Disk, John Wiley, 1994.

[23] D. Sarason, Local Dirichlet spaces as de Branges-Rovnyak spaces, Proc. Amer. Math. Soc.125 (1997), 2133–2139.

[24] B. Sz–Nagy, C. Foias, H. Bercovici, L. Kerchy: Harmonic Analysis of Operators on Hilbert

space, Springer, 2010.

Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest014700, Romania

E-mail address: Dan.Timotin@imar.ro